% \todo{
% Plan of the introduction:
% \begin{enumerate}
%  \item Biological introduction (synthetic biology $\rightarrow$ toxicology $\rightarrow$ current approach vs one that inspect causes $\rightarrow$ pathways $\rightarrow$ databases $\rightarrow$ CAD techniques (?) $\rightarrow$ need for an automatic check (?) )
%  \item Contribution: 
%    build on reaction systems, add time, model as Petri nets with timestamps, reachability analysis, upper bound on the clock 
%  \item More on technical issues? (Database confusion, ...)
% \end{enumerate}
% }


% \todo{For Franck: Can you please add the bibtex of the citations}

 \emph{Synthetic biology} \cite{synthetic} can be defined as the design and construction of new biological parts, devices, or systems for useful purposes. 
%Take, for instance, the case of drug production by  bacteria~\cite{keasling} \todo{say more...}.
%As exemplified by the drug scenario above, 
This new approach has potential effects on strategic domains such as health, energy, or environment. Thus, it necessarily  raises  a number of security issues that are mainly   related  to unexpected behaviors or reactions between the actors in the system.  
As many engineering sciences addressing sensitive technology, the risk detection and prevention must be tackled at the earliest phase of the design and pursued at each stage to guarantee the safety of the products.
As a matter of fact these problems are analyzed in the ambit of \emph{toxicology}. 

Toxicology \cite{toxicology}  studies  the adverse effects of the  exposures to chemicals at the different levels of a living entity:  organism, tissue, cell and intracellular molecular systems. In the last decade, a new field of toxicology, \emph{toxicogenomics} has emerged aiming at studying the response of the  genome to toxicants. %~\cite{synthesis-2004}. 
Most of the techniques used to address this problem are  based on empirical analysis (biomarkers, micro arrays, \dots). 
We aim at providing a more systematic way of testing synthetic products \cite{computational}.
%For instance \todo{Explain biomarkers and micro arrays...}
% In this undertaking, micro-arrays technology enables the gene expression assessment in response to specific toxic substance exposure. The  gene expression profiles trace the action mode of chemicals over dose and time characterizing patterns related to the effect of  toxic substances that identifies typical signatures of toxicity. 
%However, the signature detection mainly addresses 
This way, instead of studying the  phenomenology of the toxic impacts, we will focus on the  causes that trigger the adverse effects on organisms. Hence, we are interested in analyzing networks that represent the interactions of biological agents. Networks are classified over the nature of the interactions:  \emph{regulatory networks} describe the regulation monitoring the gene expression; \emph{signaling networks} represent the chain of signal propagation from the membrane sensors to the nucleus and \emph{metabolic networks} describes the system of chemical reactions of a cell. Altogether they form the so-called \emph{pathways} that describe  causal chains of the cellular responses to stimuli.
In particular, we believe that the main toxicity problems related to artificial pathways arise from the following two classes of situations:
\begin{enumerate}
        \item Presence/Absence of given sets of species
	\item Activation/inhibition of pathways 
\end{enumerate}
While it is clear how to connect toxicity to presence or absence of certain predefined species, the activation of undesired pathways (resp. the inhibition of desired pathways) happens when the synthetic network activates (resp. inhibits) existing/natural pathways.  Toxicity comes into play as these scenarios could break the inner equilibrium of the organism thus causing its collapse.

There is an increasing interest in computer science on how to provide tools that should support the development of all the aspects related to synthetic biology: aided design, simulation, and verification \cite{computational}.
As a matter of fact, there have been introduced a number of CAD environments~\cite{Czar2009,Bilitchenko2011} and programming languages~\cite{Basso-Blandin2012,Beal2011} that are dedicated to  synthetic biology. Independently to  these developments, several models, that span from process algebras, to Petri nets and  from rewriting systems to differential equation, have been proposed to simulate and verify properties of biological systems, for a survey see \cite{pinney,CardelliP09,journals/tcsb/Cardelli05}.

Here we propose to unify the design aspect to the verification one by proposing a tool that by taking a newly designed component verifies whether the in-silico product can be safely used in in-vivo experiments. 

\subsection{Our Contribution}
To begin our analysis, we focus on pathways. They connote the idea of  biological algorithms. In particular they are reminiscent of rewriting systems, hence a number of proposals have been introduced to be able to model, simulate and design them. P-systems \cite{DBLP:journals/ijfcs/PaunPRS11,MadhuKrithi} are among the first computing models introduced for this aim, they describe the evolution of cells through rules that are applied in a nondeterministic, maximally parallel way. 
Similarly the $\kappa$-calculus \cite{DBLP:journals/tcs/DanosL04} describes molecular biology by encoding molecules as graphs and evolution as sets of rewriting rules. 
Our work builds upon the notion of reaction systems as introduced in \cite{DBLP:journals/ijfcs/BrijderEMR11,DBLP:journals/tcs/EhrenfeuchtR07}. 
A reaction system is a set of \emph{reactions} $(R, I, P)$ where $R$ is the set of reactants, $I$ the set of inhibitors and $P$ the set of products, and  $R,I$ and $P$ are taken from a common set of species \res. Systems are based on three  foundational principles: 
\begin{enumerate}
 \item a reaction can take place only if all the reactants involved  are available but none of the inhibitors are; 
 \item if a resource is available then a sufficient amount is present to trigger a reaction  and 
 \item resources are not persistent: they become unavailable if they are not sustained by a reaction.
\end{enumerate}

We extend this model to reaction systems with duration by introducing discrete time both to species and reactions.  On one side, species could degrade, thus we associate to each entity a decay time $t$ meaning that  the resource is available and active only for an interval $t$ of time. On the other side, each reaction $(R, I, P)$ is extended with a response time $D$.
%This way, given a set of species $\res$ as before, a reaction system is a set of reactions  
%where the system has a global discrete time represented by a clock. 
Previous  principles are, then, modified in the following way: 
\begin{enumerate}
\item  a reaction of response time $D$ can take place only if all the reactants involved  are available and all  the inhibitors are unavailable during the whole reaction time;
\item if a species is present then a sufficient amount of it is available to trigger a reaction; 
\item species have a decay time governing  their presence in the system:  they are not consumed by reactions, they only degrade because of time.
\end{enumerate}

We will model the dynamics of extended reaction systems into a particular form of Petri nets with causal time \cite{DBLP:journals/entcs/ThanhKP02}.  The encoding can then be used to study the problems mentioned above. Indeed the presence (resp. absence) of a given set of species and the activation (resp. inhibition) of pathways can be reconducted to the study of reachability problems.
% \todo{Complete with discussion on reachability and the part on the bound ...}
% 
% 
% Summing up, in this paper we propose:
% \begin{enumerate}
%  \item a model of reaction systems with Petri nets with timestamps allowing directly for simulation;
%  \item an upper bound that permits to bound the net and therefore to use existing model checker tools.
% \end{enumerate}
This way we have a complete framework where problems related to the reachability of dangerous -- toxic -- states can be addressed.

\paragraph{Organization of the report.} The paper is organized as follows: Section \ref{sec:reaction} introduces the model and its encoding into Petri nets, while Section \ref{sec:concl} discusses how the toxicity problems can be tackled and concludes with some considerations on future work.

% We aim at tackling the problem of toxicogenomics from a systemic point of view: i.e. possible toxic effects should be identified  during the design phase. This raises several issues. First of all  toxic effects are strictly connected to a context of application (different organisms reacts differently to different products, or in other words toxic effects are connected to the specimen under consideration) \comment{true when talking about genes?} 
% As a matter of  fact,  biological experiments are conducted first in vitro where the effects are studied in an isolated and well known context and then  in vivo where many variables are uncontrolled and the system is subjected to unexpected  reactions.
% We aim at giving the possibility of expressing toxic properties of open systems (therefore moving directly to in vivo simulations) nevertheless the analysis should be restricted to a specific context (i.e. pathways that occur only is a specific organism) thus we need a proper way of selecting pathways of interest from a database.
% Secondly we need a proper formalism to encode pathways and perform our toxic-detection analysis. We believe that the proper formalism to do that is reaction systems as introduced in \cite{}
% 
% \comment{explain /introduce what a reaction systems is}
% 
% 
% More precisely, we assume to have a consistent database of genetic pathways where all these pathways can be encoded into reaction systems. Moreover we assume given a synthetic pathway $P$ that consist only of activation or inhibition arrows and a context of application $A$. 
% 
% Our approach consists of a four steps algorithm. The first step accounts for a proper translation of $P$ that calculates all the preconditions for the activation of the pathway.  The second step generates a proper view of the database adapted to the context $A$ and the preconditions evaluated on previous step (this accounts for the conception of a proper ontology that captures the meaning of $A$). Third steps build a reaction net taking into consideration the interplay between the synthetic pathway and the pathways selected from previous step. Finally a toxic-detection analysis can be conducted on the obtained net: i.e. reachability analysis for points 1-2, boundedness for point 3 \comment{not sure that an analysis on the boundedness of certain places can be related to fluctuation}, and cycle detection for point 4. 
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